Mastering Quadratic Equations for IGCSE Mathematics | 掌握 IGCSE 数学二次方程

📚 Mastering Quadratic Equations for IGCSE Mathematics | 掌握 IGCSE 数学二次方程

Quadratic equations are one of the most important topics in the IGCSE Mathematics syllabus. They appear in algebra, coordinate geometry, mensuration and even in statistics-based modelling questions. Building confidence with the three main solution methods will help you across many areas of the course.

二次方程是 IGCSE 数学教学大纲中最重要的主题之一。它们出现在代数、坐标几何、测量学甚至基于统计的建模问题中。熟练掌握三种主要求解方法将帮助你在课程的许多领域取得好成绩。

In this revision guide, we will cover standard form, factorising, completing the square, the quadratic formula, the discriminant, graphs and word problems. Each section gives a key idea, a worked example and a short exam tip.

在这份复习指南中，我们将涵盖标准形式、因式分解法、配方法、二次公式、判别式、图像和应用题。每一节都给出核心概念、例题和简短的考试提示。

1. What is a Quadratic Equation? | 什么是二次方程？

A quadratic equation is any equation that can be written in the standard form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. If a = 0, the x² term disappears and the equation becomes linear, so a must not be zero.

二次方程是任何可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b 和 c 是常数且 a ≠ 0。如果 a = 0，x² 项消失，方程就变成一次方程，因此 a 绝不能为零。

The word ‘quadratic’ comes from the Latin word ‘quadratus’, meaning square, because the highest power of the unknown x is 2. In IGCSE questions, you are often asked to rearrange an equation into this standard form before solving it.

“quadratic”一词源自拉丁语“quadratus”，意思是平方，因为未知数 x 的最高次数是 2。在 IGCSE 考题中，经常要求你先把方程整理成这种标准形式，然后再求解。

ax² + bx + c = 0, a ≠ 0

2. Solving by Factorising | 因式分解法求解

Factorising is usually the fastest method when a quadratic has simple integer roots. For a quadratic of the form x² + bx + c, find two numbers that multiply to give c and add to give b.

当二次方程有简单的整数根时，因式分解法通常是最快的方法。对于 x² + bx + c 型的二次式，找出两个数，使它们的乘积为 c，和为 b。

Example: Solve x² − 5x + 6 = 0. The numbers −2 and −3 multiply to 6 and add to −5, so x² − 5x + 6 = (x − 2)(x − 3). Setting each factor equal to zero gives x = 2 or x = 3.

例：解 x² − 5x + 6 = 0。数 −2 和 −3 相乘得 6，相加得 −5，因此 x² − 5x + 6 = (x − 2)(x − 3)。令每个因式为零，得 x = 2 或 x = 3。

x² − 5x + 6 = (x − 2)(x − 3) = 0 → x = 2 or x = 3

For quadratics where a ≠ 1, such as 2x² + 7x + 3, you can split the middle term or use trial factors: 2x² + 7x + 3 = (2x + 1)(x + 3).

对于 a ≠ 1 的二次式，例如 2x² + 7x + 3，你可以拆中间项或尝试因式：2x² + 7x + 3 = (2x + 1)(x + 3)。

3. Solving by Completing the Square | 配方法求解

Completing the square transforms ax² + bx + c into the form a(x + p)² + q. This method is especially useful when a quadratic cannot be factorised easily, and it leads directly to the quadratic formula.

配方法将 ax² + bx + c 转化为 a(x + p)² + q 的形式。当二次式不容易因式分解时，该方法特别有用，并且它能直接推导出二次公式。

For x² + bx, add and subtract (b/2)². Example: x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7. Setting this equal to zero gives x = −3 ± √7.

对于 x² + bx，加上并减去 (b/2)²。例：x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7。令它等于零，得 x = −3 ± √7。

x² + 6x + 2 = (x + 3)² − 7 = 0 → x = −3 ± √7

This form is also very useful for finding the minimum or maximum value of a quadratic function, because the square term is always non-negative.

这种形式对于求二次函数的最小值或最大值也非常有用，因为平方项总是非负的。

4. The Quadratic Formula | 二次公式

The quadratic formula gives the solutions of ax² + bx + c = 0 directly. Memorise it carefully and use it when factorisation is difficult or when exact surd answers are required.

二次公式直接给出 ax² + bx + c = 0 的解。请熟记该公式，在因式分解困难或需要精确根式答案时使用它。

x = (−b ± √(b² − 4ac)) / 2a

Example: Solve 2x² + 3x − 2 = 0. Here a = 2, b = 3, c = −2, so x = (−3 ± √(9 + 16)) / 4 = (−3 ± 5) / 4. The roots are x = 1/2 and x = −2.

例：解 2x² + 3x − 2 = 0。这里 a = 2，b = 3，c = −2，因此 x = (−3 ± √(9

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